On the Ishida and Du Bois complexes

Abstract

In [12] Ishida introduces a complex, denoted by $\tilde {\Omega}^{^.}_Y$, associated to a filtered semi-toroidal variety Y over Spec C and proves that it is quasi-isomorphic to the Du Bois complex $\underline{\underline{ \Omega}}^{^.}_Y$ ([5]). In this article we regard a filtered semi-toroidal variety Y as an ideally log smooth log scheme over Spec C, and we give an interpretation of the Ishida complex $\tilde {\Omega}^{^.}_Y$ in terms of logarithmic geometry. Therefore, given a log smooth log scheme X over Spec C, we use this logarithmic interpretation of the Ishida complex to construct the following distinguished triangle in the Du Bois derived category Ddiff(X): $I_M \omega^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_D \longrightarrow .$, where D = X − Xtriv (Xtriv being the trivial locus for the log structure M on X). Since the complex $I_M \omega^{^.}_X$ calculates the De Rham cohomology with compact supports of the smooth analytic space $X_{triv}^{an}$ ([20, Corollary 1.6]), this triangle is useful to give an interpretation of $H^{^.}_{DR,c}$(Xtriv/C) as the hyper-cohomology of the simple complex $\underline{\underline{s}}[\underline{\underline{\Omega}}^{^.}_X \longrightarrow \underline{\underline{\Omega}}^{^.}_D]$.